Embedded newform 111.2.q.b.59.4

• Label: 111.2.q.b.59.4
• Level: $111$
• Weight: $2$
• Character: 111.59
• Analytic conductor: $0.886$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			59.4
Character	\(\chi\)	\(=\)	111.59
Dual form			111.2.q.b.32.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.420790 - 0.902386i) q^{2} +(1.48566 - 0.890397i) q^{3} +(0.648338 - 0.772659i) q^{4} +(-1.24569 - 1.77903i) q^{5} +(-1.42863 - 0.965972i) q^{6} +(-0.867836 + 4.92174i) q^{7} +(-2.89354 - 0.775323i) q^{8} +(1.41439 - 2.64566i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{31}{36}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.420790	−	0.902386i	−0.297543	−	0.638083i	0.699524	−	0.714609i	\(-0.253395\pi\)
							−0.997068	+	0.0765252i	\(0.975617\pi\)
\(3\)	1.48566	−	0.890397i	0.857748	−	0.514071i
							\(5\)	−1.24569	−	1.77903i	−0.557090	−	0.795607i	0.437529	−	0.899204i	\(-0.355854\pi\)
−0.994619	+	0.103597i	\(0.966965\pi\)
\(7\)	−0.867836	+	4.92174i	−0.328011	+	1.86024i	0.159597	+	0.987182i	\(0.448980\pi\)
							−0.487608	+	0.873062i	\(0.662131\pi\)
\(11\)	2.36164	+	4.09049i	0.712063	+	1.23333i	0.964082	+	0.265606i	\(0.0855721\pi\)
							−0.252019	+	0.967722i	\(0.581095\pi\)
\(13\)	−0.286332	−	3.27279i	−0.0794142	−	0.907709i	−0.926510	−	0.376269i	\(-0.877207\pi\)
							0.847096	−	0.531440i	\(-0.178349\pi\)
\(17\)	0.0360699	−	0.412281i	0.00874823	−	0.0999928i	−0.990565	−	0.137042i	\(-0.956240\pi\)
							0.999313	+	0.0370495i	\(0.0117959\pi\)
\(19\)	1.79512	+	0.837079i	0.411829	+	0.192039i	0.617481	−	0.786586i	\(-0.288153\pi\)
							−0.205651	+	0.978625i	\(0.565931\pi\)
\(23\)	1.00930	+	3.76676i	0.210454	+	0.785424i	0.987718	+	0.156249i	\(0.0499403\pi\)
							−0.777264	+	0.629175i	\(0.783393\pi\)
\(29\)	−0.994311	+	3.71082i	−0.184639	+	0.689082i	0.810069	+	0.586335i	\(0.199430\pi\)
							−0.994708	+	0.102747i	\(0.967237\pi\)
\(31\)	−0.758750	+	0.758750i	−0.136275	+	0.136275i	−0.771954	−	0.635678i	\(-0.780720\pi\)
							0.635678	+	0.771954i	\(0.280720\pi\)
\(37\)	−3.19979	+	5.17314i	−0.526043	+	0.850458i
							\(41\)	0.196941	+	0.165253i	0.0307571	+	0.0258082i	0.658037	−	0.752986i	\(-0.271387\pi\)
−0.627280	+	0.778794i	\(0.715832\pi\)
\(43\)	−4.12406	−	4.12406i	−0.628913	−	0.628913i	0.318881	−	0.947795i	\(-0.396693\pi\)
							−0.947795	+	0.318881i	\(0.896693\pi\)
\(47\)	−7.62486	−	4.40221i	−1.11220	−	0.642129i	−0.172801	−	0.984957i	\(-0.555282\pi\)
							−0.939398	+	0.342828i	\(0.888615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.59.4	yes	120
3.2	odd	2	inner	111.2.q.b.59.7	yes	120
37.32	odd	36	inner	111.2.q.b.32.7	yes	120
111.32	even	36	inner	111.2.q.b.32.4	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.32.4	✓	120	111.32	even	36	inner
111.2.q.b.32.7	yes	120	37.32	odd	36	inner
111.2.q.b.59.4	yes	120	1.1	even	1	trivial
111.2.q.b.59.7	yes	120	3.2	odd	2	inner